Linear Algebra and Its Applications (5th Edition)

by
Lay, David C.; Lay, Steven R.; McDonald, Judi J.

Chapter 1 - Linear Equations in Linear Algebra - 1.8 Exercises: 5

Answer

$\begin{bmatrix}3-3k\\1-2k\\k\end{bmatrix}$ for any real number $k$ (not unique)

Work Step by Step

Finding the vector with a given image under some matrix transformation is equivalent to finding the solution set of a linear system. Hence, we begin with the augmented matrix:
$\begin{bmatrix}1&-5&-7&-2\\-3&7&5&-2\end{bmatrix}$
Add $3$ times row (1) to row (2):
$\begin{bmatrix}1&-5&-7&-2\\0&-8&-16&-8\end{bmatrix}$
Multiply row (2) by $-\frac{1}{8}$:
$\begin{bmatrix}1&-5&-7&-2\\0&1&2&1\end{bmatrix}$
Add $5$ times row (2) to row (1):
$\begin{bmatrix}1&0&3&3\\0&1&2&1\end{bmatrix}$
Since the third column contains no pivot, the solution is not unique, and we have $\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}=\begin{bmatrix}3-3k\\1-2k\\k\end{bmatrix}$.